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%I #4 Jan 15 2018 08:56:03
%S 11,12,13,14,15,16,17,21,22,23,24,25,26,31,32,33,34,35,41,42,43,44,51,
%T 52,53,61,62,71,99,100,102,103,104,105,106,107,108,109,110,112,113,
%U 114,115,116,117,118,119,120,122,123,124,125,126,127,128,129,130
%N Numbers having an up-first zigzag pattern in base 9; see Comments.
%C A number n having base-b digits d(m), d(m-1),..., d(0) such that d(i) != d(i+1) for 0 <= i < m shows a zigzag pattern of one or more segments, in the following sense. Writing U for up and D for down, there are two kinds of patterns: U, UD, UDU, UDUD, ... and D, DU, DUD, DUDU, ... . In the former case, we say n has an "up-first zigzag pattern in base b"; in the latter, a "down-first zigzag pattern in base b". Example: 2,4,5,3,0,1,4,2 has segments 2,4,5; 5,3,0; 0,1,4; and 4,2, so that 24530142, with pattern UDUD, has an up-first zigzag pattern in base 10, whereas 4,2,5,3,0,1,4,2 has a down-first pattern. The sequences A297143-A297145 partition the natural numbers. See the guide at A297146.
%e Base-9 digits of 10000: 1,4,6,4,1, with pattern UD, so that 10000 is in the sequence.
%t a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t b = 9; t = Table[a[n, b], {n, 1, 10*z}];
%t u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &] (* A297143 *)
%t v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &] (* A297144 *)
%t Complement[Range[z], Union[u, v]] (* A297145 *)
%Y Cf. A297144, A297145.
%K nonn,easy,base
%O 1,1
%A _Clark Kimberling_, Jan 15 2018
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